Introduction

There have been many attempts in the last decades to arrive at a theory of noncommutative geometry applicable to ‘coordinate’ algebras that are not necessarily commutative, notably that of A. Connes coming out of abstract C*-algebra theory in the light of the Gelfand-Naimark and Serre-Swan theorems. One has tools such as cyclic cohomology and examples such as the noncommutative torus and other foliation C*-algebras. Another ‘bottom up’ approach, which we outline, is based on the idea that the theory should be guided by the inclusion of the large vein of ‘naturally occuring’ examples, the coordinate algebras of the quantum groups Uq(g) in particular, and Hopf algebras in general, whose validity for several branches of mathematics has already been established. This is similar to the key role that Lie groups played in the development of modern differential geometry. Much progress has been made in recent years and there is by now (at least at the algebraic level) a more or less clear formulation of ‘quantum manifold’ suggested by this approach. After being validated on the q-deformation examples such as quantum groups, quantum homogeneous spaces etc, one can eventually apply the theory quite broadly to a wide range of unital algebras.